Theorem blssex 15457

Description: Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)

Assertion

Ref	Expression
blssex	|- ( ( D e. ( *Met ` X ) /\ P e. X ) -> ( E. x e. ran ( ball ` D ) ( P e. x /\ x C_ A ) <-> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )

Distinct variable groups:   x, r, A   D, r, x   P, r, x   X, r, x

Proof of Theorem blssex

Step	Hyp	Ref	Expression
1		blss 15455	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ x e. ran ( ball ` D ) /\ P e. x ) -> E. r e. RR+ ( P ( ball ` D ) r ) C_ x )
2		sstr 3256	. . . . . . . . 9 |- ( ( ( P ( ball ` D ) r ) C_ x /\ x C_ A ) -> ( P ( ball ` D ) r ) C_ A )
3	2	expcom 116	. . . . . . . 8 |- ( x C_ A -> ( ( P ( ball ` D ) r ) C_ x -> ( P ( ball ` D ) r ) C_ A ) )
4	3	reximdv 2651	. . . . . . 7 |- ( x C_ A -> ( E. r e. RR+ ( P ( ball ` D ) r ) C_ x -> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )
5	1, 4	syl5com 29	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ x e. ran ( ball ` D ) /\ P e. x ) -> ( x C_ A -> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )
6	5	3expa 1234	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ x e. ran ( ball ` D ) ) /\ P e. x ) -> ( x C_ A -> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )
7	6	expimpd 363	. . . 4 |- ( ( D e. ( *Met ` X ) /\ x e. ran ( ball ` D ) ) -> ( ( P e. x /\ x C_ A ) -> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )
8	7	adantlr 481	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ x e. ran ( ball ` D ) ) -> ( ( P e. x /\ x C_ A ) -> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )
9	8	rexlimdva 2668	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X ) -> ( E. x e. ran ( ball ` D ) ( P e. x /\ x C_ A ) -> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )
10		simpll 531	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> D e. ( *Met ` X ) )
11		simplr 533	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> P e. X )
12		rpxr 10044	. . . . . 6 |- ( r e. RR+ -> r e. RR* )
13	12	ad2antrl 494	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> r e. RR* )
14		blelrn 15447	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ r e. RR* ) -> ( P ( ball ` D ) r ) e. ran ( ball ` D ) )
15	10, 11, 13, 14	syl3anc 1278	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> ( P ( ball ` D ) r ) e. ran ( ball ` D ) )
16		simprl 535	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> r e. RR+ )
17		blcntr 15443	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ r e. RR+ ) -> P e. ( P ( ball ` D ) r ) )
18	10, 11, 16, 17	syl3anc 1278	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> P e. ( P ( ball ` D ) r ) )
19		simprr 537	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> ( P ( ball ` D ) r ) C_ A )
20		eleq2 2302	. . . . . 6 |- ( x = ( P ( ball ` D ) r ) -> ( P e. x <-> P e. ( P ( ball ` D ) r ) ) )
21		sseq1 3271	. . . . . 6 |- ( x = ( P ( ball ` D ) r ) -> ( x C_ A <-> ( P ( ball ` D ) r ) C_ A ) )
22	20, 21	anbi12d 477	. . . . 5 |- ( x = ( P ( ball ` D ) r ) -> ( ( P e. x /\ x C_ A ) <-> ( P e. ( P ( ball ` D ) r ) /\ ( P ( ball ` D ) r ) C_ A ) ) )
23	22	rspcev 2929	. . . 4 |- ( ( ( P ( ball ` D ) r ) e. ran ( ball ` D ) /\ ( P e. ( P ( ball ` D ) r ) /\ ( P ( ball ` D ) r ) C_ A ) ) -> E. x e. ran ( ball ` D ) ( P e. x /\ x C_ A ) )
24	15, 18, 19, 23	syl12anc 1276	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> E. x e. ran ( ball ` D ) ( P e. x /\ x C_ A ) )
25	24	rexlimdvaa 2669	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X ) -> ( E. r e. RR+ ( P ( ball ` D ) r ) C_ A -> E. x e. ran ( ball ` D ) ( P e. x /\ x C_ A ) ) )
26	9, 25	impbid 129	1 |- ( ( D e. ( *Met ` X ) /\ P e. X ) -> ( E. x e. ran ( ball ` D ) ( P e. x /\ x C_ A ) <-> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   C_ wss 3220   ran crn 4773   ` cfv 5375  (class class class)co 6078   RR*cxr 8352   RR+crp 10036   *Metcxmet 14848   ballcbl 14850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8263  ax-resscn 8264  ax-1cn 8265  ax-1re 8266  ax-icn 8267  ax-addcl 8268  ax-addrcl 8269  ax-mulcl 8270  ax-mulrcl 8271  ax-addcom 8272  ax-mulcom 8273  ax-addass 8274  ax-mulass 8275  ax-distr 8276  ax-i2m1 8277  ax-0lt1 8278  ax-1rid 8279  ax-0id 8280  ax-rnegex 8281  ax-precex 8282  ax-cnre 8283  ax-pre-ltirr 8284  ax-pre-ltwlin 8285  ax-pre-lttrn 8286  ax-pre-apti 8287  ax-pre-ltadd 8288  ax-pre-mulgt0 8289  ax-pre-mulext 8290  ax-arch 8291

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-po 4439  df-iso 4440  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fv 5383  df-riota 6031  df-ov 6081  df-oprab 6082  df-mpo 6083  df-1st 6367  df-2nd 6368  df-map 6917  df-pnf 8355  df-mnf 8356  df-xr 8357  df-ltxr 8358  df-le 8359  df-sub 8492  df-neg 8493  df-reap 8896  df-ap 8903  df-div 8996  df-inn 9287  df-2 9345  df-n0 9546  df-z 9627  df-uz 9904  df-q 10002  df-rp 10037  df-xneg 10156  df-xadd 10157  df-psmet 14855  df-xmet 14856  df-bl 14858

This theorem is referenced by:  blbas  15460  elmopn2  15476  mopni2  15510  metss  15521  tgioo  15581